北京大学：量子多体的理论与计算（讲义）Quantum Theory of Many Particle Systems

QuantumTheory of ManyParticleSystems施均仁(junrenshi@pku.edu.cn)March 28,2023

Quantum Theory of Many Particle Systems 施均仁 (junrenshi@pku.edu.cn) March 28, 2023

Contents11 Second quantization and coherent statesNO51.111.1Quantum mechanicsNO52.121.2QuantumstatisticalmechanicsNO51.231.3Identical particles41.4Creation and annihilation operatorsNO51.441.4.1Basics.......71.4.2Secondquantized Hamiltonians9NO51.51.5Coherent states.91.5.1Boson coherent states101.5.2Grassmann algebra1.5.3Fermion coherent states .111.5.413Gaussian integrals1.613Summary152Green'sfunctions152.1 Green'sfunctions and observables.152.1.1Time-ordered Green'sfunctionsNO55.1172.1.2Evaluation ofObservablesNO52.1182.1.3Response functions.HJ53.2202.1.4Species of Green's functions2.2:21Fluctuation-dissipationtheoremHJ53.32.2.121Real timeGreen'sfunctionsFW5312.2.2Thermal Green's function and analytic continuation22HJ542.322Non-equilibrium Green's function242.41 Summary25NO52.23Functional integrals253.1 Feynman path integrals.263.2Imaginary time path integrals and the partition function. .3.328Functional integrals.293.4PartitionfunctionandGreen'sfunctionsi

Contents 1 Second quantization and coherent states 1 1.1 Quantum mechanics NO§1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Quantum statistical mechanics NO§2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Identical particles NO§1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.1 Basics NO§1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.2 Second quantized Hamiltonians . . . . . . . . . . . . . . . . . . . . . 7 1.5 Coherent states NO§1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.1 Boson coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.2 Grassmann algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5.3 Fermion coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.4 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Green’s functions 15 2.1 Green’s functions and observables . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Time-ordered Green’s functions . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 Evaluation of Observables NO§5.1 . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.3 Response functions NO§2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.4 Species of Green’s functions HJ§3.2 . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Fluctuation-dissipation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1 Real time Green’s functions HJ§3.3 . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.2 Thermal Green’s function and analytic continuation FW§31 . . . . . . . . 22 2.3 Non-equilibrium Green’s function HJ§4 . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Functional integrals NO§2.2 25 3.1 Feynman path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Imaginary time path integrals and the partition function . . . . . . . . . . 26 3.3 Functional integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 Partition function and Green’s functions . . . . . . . . . . . . . . . . . . . . 29 ii

32NO52.34 PerturbationtheoryNO52.3：324.1General strategy4.2 Finite temperature formalism34：344.2.1Labeled Feynman diagrams.364.2.2UnlabeledFeynmanDiagrams4.2.336Hugenholtz diagrams374.2.4Frequencyandmomentumrepresentation394.2.5Thelinked clustertheorem.394.2.6Green'sfunctionsNO53.14.3Zero temperatureformalism41414.3.1Ground state energy and Green's function434.3.2Diagram Rules4.3.344FreeFermion propagatorsHJ 54.3454.4Contourformalism454.5Summary495Effectiveactiontheoryandenergyfunctionals495.1Effectiveaction..NO52.4525.2Irreducible diagrams and integral equations525.2.1Self-energy and Dyson's equation.535.2.2Perturbative construction of the Luttinger-Ward functional545.2.3Second ordervertexfunction555.2.4Higher order equationsHJ55.25.2.556KeldyshFormulation5.2.656Other sourcesGV58575.3Landau Fermi-liquid theoryGV58.3575.3.1PhenomenologicalapproachGV$8.5595.3.2Microscopic underpinning5.461Generalizations636 TheoryofelectronliquidFW512636.1Energy....636.1.1Hartree-Fock approximation.：656.1.2High order contributions6.1.366General structure of the self-energy:676.2Densityresponsefunction.GV 53.3676.2.1Basic propertiesGV 55.36.2.269Random phase approximation (RPA)GV $5.4726.2.3Local field correction756.3Plasmon...GV 55.3.3756.3.1CollectiveexcitationAS 56.26.3.277Functional integralsofplasmonsAS 56.2796.3.3Collective excitationsii

4 Perturbation theory NO§2.3 32 4.1 General strategy NO§2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 Finite temperature formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.2.1 Labeled Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . 34 4.2.2 Unlabeled Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . 36 4.2.3 Hugenholtz diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2.4 Frequency and momentum representation . . . . . . . . . . . . . . 37 4.2.5 The linked cluster theorem . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2.6 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3 Zero temperature formalism NO§3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3.1 Ground state energy and Green’s function . . . . . . . . . . . . . . 41 4.3.2 Diagram Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.3 Free Fermion propagators . . . . . . . . . . . . . . . . . . . . . . . . 44 4.4 Contour formalism HJ §4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 Effective action theory and energy functionals 49 5.1 Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Irreducible diagrams and integral equations NO§2.4 . . . . . . . . . . . . . . . . . 52 5.2.1 Self-energy and Dyson’s equation . . . . . . . . . . . . . . . . . . . . 52 5.2.2 Perturbative construction of the Luttinger-Ward functional . . . . 53 5.2.3 Second order vertex function . . . . . . . . . . . . . . . . . . . . . . 54 5.2.4 Higher order equations . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2.5 Keldysh Formulation HJ§5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2.6 Other sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.3 Landau Fermi-liquid theory GV§8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.3.1 Phenomenological approach GV§8.3 . . . . . . . . . . . . . . . . . . . . . . . 57 5.3.2 Microscopic underpinning GV§8.5 . . . . . . . . . . . . . . . . . . . . . . . . 59 5.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6 Theory of electron liquid 63 6.1 Energy FW§12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.1.1 Hartree-Fock approximation . . . . . . . . . . . . . . . . . . . . . . . 63 6.1.2 High order contributions . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.1.3 General structure of the self-energy . . . . . . . . . . . . . . . . . . 66 6.2 Density response function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.2.1 Basic properties GV §3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.2.2 Random phase approximation (RPA) GV §5.3 . . . . . . . . . . . . . . . . . . 69 6.2.3 Local field correction GV §5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.3 Plasmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.3.1 Collective excitation GV §5.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.3.2 Functional integrals of plasmons AS §6.2 . . . . . . . . . . . . . . . . . . . . 77 6.3.3 Collective excitations AS §6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 iii

817Phasetransitions and spontaneous symmetrybreaking817.1Generaltheory.....NO 54.1817.1.1PhasetransitionsNO 54.1, 4.27.1.2Landautheory81NO 54.37.1.3Meanfieldtheory84NO 54.47.1.4Fluctuations8587AS 56.37.2Bose-Einstein condensation and superfluidity7.2.187Phasetransition.887.2.2Superfluidity897.2.3BogoliubovtransformationAS 56.4907.3Superconductivity7.3.190Introduction917.3.2Cooper instability.7.3.3Mean field theory93957.3.4EffectivefieldtheoryandAnderson-Higgsmechanismiv

7 Phase transitions and spontaneous symmetry breaking 81 7.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.1.1 Phase transitions NO §4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.1.2 Landau theory NO §4.1, 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.1.3 Mean field theory NO §4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.1.4 Fluctuations NO §4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.2 Bose-Einstein condensation and superfluidity AS §6.3 . . . . . . . . . . . . . . . . . 87 7.2.1 Phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.2.2 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.2.3 Bogoliubov transformation . . . . . . . . . . . . . . . . . . . . . . . . 89 7.3 Superconductivity AS §6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.3.2 Cooper instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.3.3 Mean field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.3.4 Effective field theory and Anderson-Higgs mechanism . . . . . . . 95 iv

PrefaceOverview:Anintroductorycourseforthequantumtheoryofmany-bodysystems;·Asurvey of general principles and language;:Anoverviewonmanybodytechniques;.Bridging the gap between thequantum many-body theory and real material calculations;Contents.BasicTheory-Secondquantizationandcoherentstates-Green's functions-Functionalintegralformalism-Perturbationtheory-Effectiveactiontheory:Applicationstophysical systems-Theoryofelectronliquids-BrokensymmetryandphasetransitionsV

Preface Overview • An introductory course for the quantum theory of many-body systems; • A survey of general principles and language; • An overview on many body techniques; • Bridging the gap between the quantum many-body theory and real material calculations; Contents • Basic Theory – Second quantization and coherent states – Green’s functions – Functional integral formalism – Perturbation theory – Effective action theory • Applications to physical systems – Theory of electron liquids – Broken symmetry and phase transitions v