Theoretical physics 9 - Fundamentals of many-body physics

701 Pages·2018·4.587 MB·English

by  Wolfgang Nolting

Theoretical Physics 9
Wolfgang Nolting
Theoretical Physics 9
Fundamentals of Many-body Physics
Second Edition
Translated by William D. Brewer
123
WolfgangNolting
Humboldt-UniversitätBerlinInst.Physik
Berlin,Germany
Translator
WilliamD.Brewer
FUBerlin
FBPhysik
Inst.f.Experimentalphysik
Berlin,Germany
ISBN978-3-319-98324-0 ISBN978-3-319-98326-4 (eBook)
https://doi.org/10.1007/978-3-319-98326-4
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1stedition:©Springer-VerlagBerlinHeidelberg2009
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Preface
The goal of the present course on “Fundamentals of Theoretical Physics” is to
be a direct accompaniment to the lower-division study of physics, and it aims
at providing the physical tools in the most straightforward and compact form as
needed by the students in order to master theoretically more complex topics and
problemsinadvancedstudiesandinresearch.Thepresentationisthusintentionally
designed to be sufficiently detailed and self-contained – sometimes, admittedly,
at the cost of a certain elegance – to permit individual study without reference to
thesecondaryliterature.Thisvolumedealswiththequantumtheoryofmany-body
systems.Buildinguponabasicknowledgeofquantummechanicsandofstatistical
physics,moderntechniquesforthedescriptionofinteractingmany-particlesystems
aredevelopedandappliedtovariousrealproblems,mainlyfromtheareaofsolid-
state physics. A thorough revision should guarantee that the reader can access the
relevant research literature without experiencing major problems in terms of the
conceptsandvocabulary,techniquesanddeductivemethodsfoundthere.
The world which surrounds us consists of very many particles interacting
with one another, and their description requires in principle the solution of
a corresponding number of coupled quantum-mechanical equations of motion
(Schrödinger equations), which, however, is possible only in exceptional cases in
amathematicallystrictsense.Theconceptsofelementaryquantummechanicsand
quantumstatisticsarethereforenotdirectlyapplicableintheforminwhichwehave
thus far encountered them. They require an extension and restructuring, which is
termed“many-bodytheory”.
Firstofall,wehavetolookforpossibilitiesforformulatingrealmany-bodyprob-
lemsinamathematicallycorrectbutstillmanageableway.Ifthesystemsconsidered
arecomposedofdistinguishableparticles,theirdescriptioncanbeobtaineddirectly
fromthegeneralpostulatesofquantummechanics.Moreinteresting,however,are
systemsofidenticalparticles,whoseN-particlewavefunctionsmustfulfilquitespe-
cialsymmetryrequirements.Workingdirectlywiththerequired(anti-)symmetrised
wavefunctionsprovestobeextraordinarilytedious.Afirstperceptiblesimplification
is provided in this connection by the formalism of second quantisation. It allows
a quite elegant description but of course does not provide an actual solution to
v
vi Preface
the problem. The student who has been confronted in lower-division courses with
problems which as a rule can be treated with mathematical rigour has to become
accustomedtotheideathatrealisticmany-bodyproblemscanpracticallyneverbe
treatedexactly.Inordertoneverthelessfulfilthecentralfunctionofatheoretician,
i.e.thedescriptionandexplanationofexperiments,someconcessionsmustbemade.
This includes, as a first step, the construction of a theoretical model which can
be understood as a caricature of the real world, in which nonessential details are
suppressed and only the essence of the problem is emphasised. Finding such a
theoreticalmodelmustbeconsideredtobeanontrivialchallengefortheoreticians.
Chapter 2 therefore treats the formulation and justification of important standard
modelsoftheoreticalphysicsindetail.Theirpresentationiscarriedoutconsistently
usingtheformalismofsecondquantisationfromChap.1.
Unfortunately, the real situation can seldom be caricatured in such a way that
the resulting model is on the one hand still sufficiently realistic and on the other
canbetreatedwithmathematicalrigour.Thus,oneusuallyhastoacceptadditional
approximationsinordertofindsolutions.Apowerfultechniqueinthisconnection
has proven to be the Green’s function method, with its concept of quasi-particles.
TheabstracttheoryisdiscussedinChap.3andthenappliedtonumerousconcrete
problemsinChap.4.DiagrammaticmethodsofsolutionareworkedoutinChaps.5
and 6. They should be included nowadays within the indispensable repertoire of
every theoretician. A number of exercises (together with their explicit solutions)
are also included in this volume and are in particular designed to help the student
to acquire a facility for working with the formalism and applying it to concrete
topics.Thesolutionsgiven,however,shouldnottemptthereadertoforbearmaking
a serious effort to solve the problems independently. At the end of each major
chapter, questions are included, which can be useful to test the knowledge gained
bythereaderandinpreparingforexaminations.
This book is the result of diverse special-topics lecture courses on many-body
theorywhichIhavegivenattheuniversitiesofWürzburg,Münster,Osnabrück,and
Berlin(Germany),Warangal(India),Valladolid(Spain),Irbid(Jordan)andHarbin
(China). I am very grateful to the students of those courses for their constructive
criticism. It is quite clear to me that the material in this volume with certainty
no longer belongs to lower-division physics. However, I also believe that it is
indispensable for making the transition to independent research as a theoretician.
Since the available textbook literature on the subject of many-body theory as a
rulepresupposesadvancedknowledgeandsubstantialexperienceonthepartofthe
reader,thepresentbookmight–hopefully–beveryusefulforthe“beginner”.Iam
very grateful to the Springer-Verlag for their concurring assessment as well as for
theirprofessionalcooperation.
Berlin,Germany WolfgangNolting
August2008
Contents
1 SecondQuantisation......................................................... 1
1.1 IdenticalParticles ....................................................... 2
1.2 The“Continuous”FockRepresentation ............................... 7
1.3 The“Discrete”FockRepresentation................................... 21
1.4 Exercises................................................................. 28
1.5 Self-ExaminationQuestions............................................ 34
1.5.1 ForSect.1.1.................................................... 34
1.5.2 ForSect.1.2.................................................... 34
1.5.3 ForSect.1.3.................................................... 35
2 Many-BodyModelSystems................................................. 37
2.1 CrystalElectrons........................................................ 39
2.1.1 Non-interactingBlochElectrons ............................. 39
2.1.2 TheJelliumModel ............................................ 44
2.1.3 TheHubbardModel........................................... 56
2.1.4 Exercises....................................................... 60
2.2 LatticeVibrations ....................................................... 65
2.2.1 TheHarmonicApproximation................................ 65
2.2.2 ThePhononGas............................................... 70
2.2.3 Exercises....................................................... 76
2.3 TheElectron-PhononInteraction....................................... 78
2.3.1 TheHamiltonian............................................... 78
2.3.2 TheEffectiveElectron-ElectronInteraction ................. 82
2.3.3 Exercises....................................................... 86
2.4 SpinWaves.............................................................. 90
2.4.1 ClassificationofMagneticSolids............................. 90
2.4.2 ModelConcepts ............................................... 92
2.4.3 Magnons....................................................... 95
2.4.4 TheSpin-WaveApproximation............................... 100
2.4.5 Exercises....................................................... 102
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viii Contents
2.5 Self-ExaminationQuestions............................................ 105
2.5.1 ForSect.2.1.................................................... 105
2.5.2 ForSect.2.2.................................................... 106
2.5.3 ForSect.2.3.................................................... 107
2.5.4 ForSect.2.4.................................................... 107
3 Green’sFunctions............................................................ 109
3.1 PreliminaryConsiderations............................................. 109
3.1.1 Representations................................................ 109
3.1.2 Linear-ResponseTheory...................................... 116
3.1.3 TheMagneticSusceptibility.................................. 120
3.1.4 TheElectricalConductivity................................... 122
3.1.5 TheDielectricFunction....................................... 125
3.1.6 Spectroscopies,SpectralDensity............................. 127
3.1.7 Exercises....................................................... 133
3.2 Double-TimeGreen’sFunctions ....................................... 135
3.2.1 EquationsofMotion .......................................... 135
3.2.2 SpectralRepresentations...................................... 140
3.2.3 TheSpectralTheorem......................................... 145
3.2.4 ExactExpressions............................................. 148
3.2.5 TheKramers-KronigRelations............................... 152
3.2.6 Exercises....................................................... 154
3.3 FirstApplications ....................................................... 157
3.3.1 Non-interactingBlochElectrons ............................. 157
3.3.2 FreeSpinWaves............................................... 163
3.3.3 TheTwo-SpinProblem........................................ 166
3.3.4 Exercises....................................................... 178
3.4 TheQuasi-particleConcept............................................. 181
3.4.1 One-ElectronGreen’sFunctions.............................. 181
3.4.2 TheElectronicSelf-Energy................................... 184
3.4.3 Quasi-particles................................................. 189
3.4.4 Quasi-particleDensityofStates.............................. 194
3.4.5 InternalEnergy................................................ 196
3.4.6 Exercises....................................................... 199
3.5 Self-ExaminationQuestions............................................ 200
3.5.1 ForSect.3.1.................................................... 200
3.5.2 ForSect.3.2.................................................... 201
3.5.3 ForSect.3.3.................................................... 201
3.5.4 ForSect.3.4.................................................... 202
4 SystemsofInteractingParticles............................................ 205
4.1 ElectronsinSolids ...................................................... 205
4.1.1 TheLimitingCaseofanInfinitelyNarrowBand............ 205
4.1.2 TheHartree-FockApproximation............................ 209
4.1.3 ElectronicCorrelations........................................ 214
4.1.4 TheInterpolationMethod..................................... 217
Contents ix
4.1.5 TheMethodofMoments...................................... 219
4.1.6 TheExactlyHalf-FilledBand ................................ 228
4.1.7 Exercises....................................................... 232
4.2 CollectiveElectronicExcitations....................................... 236
4.2.1 ChargeScreening(Thomas-FermiApproximation)......... 237
4.2.2 ChargeDensityWaves,Plasmons ............................ 241
4.2.3 SpinDensityWaves,Magnons ............................... 250
4.2.4 Exercises....................................................... 254
4.3 ElementaryExcitationsinDisorderedAlloys ......................... 257
4.3.1 FormulationoftheProblem................................... 257
4.3.2 TheEffective-MediumMethod............................... 261
4.3.3 TheCoherentPotentialApproximation...................... 263
4.3.4 DiagrammaticMethods ....................................... 267
4.3.5 Applications ................................................... 278
4.4 SpinSystems............................................................ 280
4.4.1 TheTyablikowApproximation............................... 280
4.4.2 “Renormalised”SpinWaves.................................. 288
4.4.3 Exercises....................................................... 293
4.5 TheElectron-MagnonInteraction...................................... 294
4.5.1 Magnetic4f Systems(s-f-Model)............................ 295
4.5.2 TheInfinitelyNarrowBand................................... 297
4.5.3 TheAlloyAnalogy............................................ 303
4.5.4 TheMagneticPolaron......................................... 305
4.5.5 Exercises....................................................... 314
4.6 Self-ExaminationQuestions............................................ 316
4.6.1 ForSect.4.1.................................................... 316
4.6.2 ForSect.4.2.................................................... 317
4.6.3 ForSect.4.3.................................................... 318
4.6.4 ForSect.4.4.................................................... 318
4.6.5 ForSect.4.5.................................................... 319
5 PerturbationTheory(T =0)............................................... 321
5.1 CausalGreen’sFunctions............................................... 321
5.1.1 “Conventional”Time-DependentPerturbationTheory...... 321
5.1.2 “Switchingon”theInteractionAdiabatically................ 326
5.1.3 CausalGreen’sFunctions..................................... 332
5.1.4 Exercises....................................................... 336
5.2 Wick’sTheorem......................................................... 337
5.2.1 TheNormalProduct........................................... 337
5.2.2 Wick’sTheorem............................................... 341
5.2.3 Exercises....................................................... 346
5.3 FeynmanDiagrams ..................................................... 347
5.3.1 PerturbationExpansionfortheVacuumAmplitude......... 347
5.3.2 TheLinked-ClusterTheorem................................. 356
5.3.3 ThePrincipalTheoremofConnectedDiagrams............. 361
5.3.4 Exercises....................................................... 364
x Contents
5.4 Single-ParticleGreen’sFunctions...................................... 365
5.4.1 DiagrammaticPerturbationExpansions...................... 365
5.4.2 TheDysonEquation........................................... 371
5.4.3 Exercises....................................................... 375
5.5 TheGround-StateEnergyoftheElectronGas(JelliumModel)...... 376
5.5.1 First-OrderPerturbationTheory.............................. 376
5.5.2 Second-OrderPerturbationTheory........................... 379
5.5.3 TheCorrelationEnergy....................................... 385
5.6 DiagrammaticPartialSums............................................. 397
5.6.1 ThePolarisationPropagator .................................. 397
5.6.2 EffectiveInteractions.......................................... 403
5.6.3 VertexFunction................................................ 408
5.6.4 Exercises....................................................... 411
5.7 Self-ExaminationQuestions............................................ 413
5.7.1 ForSect.5.1.................................................... 413
5.7.2 ForSect.5.2.................................................... 413
5.7.3 ForSect.5.3.................................................... 414
5.7.4 ForSect.5.4.................................................... 414
5.7.5 ForSect.5.5.................................................... 414
5.7.6 ForSect.5.6.................................................... 415
6 PerturbationTheoryatFiniteTemperatures............................. 417
6.1 TheMatsubaraMethod................................................. 417
6.1.1 MatsubaraFunctions.......................................... 418
6.1.2 TheGrandCanonicalPartitionFunction..................... 424
6.1.3 TheSingle-ParticleMatsubaraFunction..................... 427
6.1.4 Exercises....................................................... 431
6.2 DiagrammaticPerturbationTheory .................................... 432
6.2.1 Wick’sTheorem............................................... 432
6.2.2 DiagramAnalysisoftheGrand-CanonicalPartition
Function........................................................ 436
6.2.3 RingDiagrams................................................. 443
6.2.4 Single-ParticleMatsubaraFunctions......................... 446
6.2.5 TheDysonEquationandSkeletonDiagrams................ 451
6.2.6 TheHartree-FockApproximation............................ 456
6.2.7 Second-Order“PerturbationTheory” ........................ 457
6.2.8 TheHubbardModel........................................... 460
6.2.9 TheJelliumModel ............................................ 462
6.2.10 The Imaginary Part of the Self Energy in the
Low-EnergyRegion........................................... 464
6.2.11 Quasi-particlesandtheFermiLiquid......................... 467
6.2.12 Exercises....................................................... 474
6.3 Two-ParticleMatsubaraFunctions..................................... 477
6.3.1 DensityCorrelation............................................ 477
6.3.2 ThePolarisationPropagator .................................. 485